Lecture 2. Matrix Operations. • transpose, sum & difference, scalar multiplication.
• matrix multiplication, matrix-vector product. • matrix inverse. 2–1 ...
Lecture 2 Matrix Operations
• transpose, sum & difference, scalar multiplication • matrix multiplication, matrix-vector product • matrix inverse
2–1
Matrix transpose transpose of m × n matrix A, denoted AT or A′, is n × m matrix with A
T
ij
= Aji
rows and columns of A are transposed in AT T 0 4 0 7 3 . example: 7 0 = 4 0 1 3 1 • transpose converts row vectors to column vectors, vice versa • A
T T
=A
Matrix Operations
2–2
Matrix addition & subtraction if A and B are both m × n, 1 0 4 example: 7 0 + 2 0 3 1
we form A + B by adding corresponding entries 1 6 2 3 = 9 3 3 5 4
can add row or column vectors same way (but never to each other!) 1 6 0 6 matrix subtraction is similar: −I = 9 3 9 2 (here we had to figure out that I must be 2 × 2)
Matrix Operations
2–3
Properties of matrix addition • commutative: A + B = B + A • associative: (A + B) + C = A + (B + C), so we can write as A + B + C • A + 0 = 0 + A = A; A − A = 0 • (A + B)T = AT + B T
Matrix Operations
2–4
Scalar multiplication we can multiply a number (a.k.a. scalar ) by a matrix by multiplying every entry of the matrix by the scalar this is denoted by juxtaposition or ·, with the scalar on the left: −2 −12 1 6 (−2) 9 3 = −18 −6 −12 0 6 0 (sometimes you see scalar multiplication with the scalar on the right) • (α + β)A = αA + βA; (αβ)A = (α)(βA) • α(A + B) = αA + αB • 0 · A = 0; 1 · A = A Matrix Operations
2–5
Matrix multiplication if A is m × p and B is p × n we can form C = AB, which is m × n Cij =
p X
aik bkj = ai1b1j + · · · + aipbpj ,
i = 1, . . . , m,
j = 1, . . . , n
k=1
to form AB, #cols of A must equal #rows of B; called compatible • to find i, j entry of the product C = AB, you need the ith row of A and the jth column of B • form product of corresponding entries, e.g., third component of ith row of A and third component of jth column of B • add up all the products Matrix Operations
2–6
Examples example 1:
1 6 9 3
0 −1 −1 2
=
−6 11 −3 −3
for example, to get 1, 1 entry of product: C11 = A11B11 + A12B21 = (1)(0) + (6)(−1) = −6
example 2:
0 −1 −1 2
1 6 9 3
=
−9 −3 17 0
these examples illustrate that matrix multiplication is not (in general) commutative: we don’t (always) have AB = BA Matrix Operations
2–7
Properties of matrix multiplication • 0A = 0, A0 = 0 (here 0 can be scalar, or a compatible matrix) • IA = A, AI = A • (AB)C = A(BC), so we can write as ABC • α(AB) = (αA)B, where α is a scalar • A(B + C) = AB + AC, (A + B)C = AC + BC • (AB)T = B T AT
Matrix Operations
2–8
Matrix-vector product very important special case of matrix multiplication: y = Ax • A is an m × n matrix • x is an n-vector • y is an m-vector
yi = Ai1x1 + · · · + Ainxn,
i = 1, . . . , m
can think of y = Ax as • a function that transforms n-vectors into m-vectors • a set of m linear equations relating x to y Matrix Operations
2–9
Inner product if v is a row n-vector and w is a column n-vector, then vw makes sense, and has size 1 × 1, i.e., is a scalar: vw = v1w1 + · · · + vnwn if x and y are n-vectors, xT y is a scalar called inner product or dot product of x, y, and denoted hx, yi or x · y: hx, yi = xT y = x1y1 + · · · + xnyn (the symbol · can be ambiguous — it can mean dot product, or ordinary matrix product) Matrix Operations
2–10
Matrix powers if matrix A is square, then product AA makes sense, and is denoted A2 more generally, k copies of A multiplied together gives Ak : Ak = A · · · A} | A {z k
by convention we set A0 = I (non-integer powers like A1/2 are tricky — that’s an advanced topic) we have Ak Al = Ak+l
Matrix Operations
2–11
Matrix inverse if A is square, and (square) matrix F satisfies F A = I, then • F is called the inverse of A, and is denoted A−1 • the matrix A is called invertible or nonsingular if A doesn’t have an inverse, it’s called singular or noninvertible by definition, A−1A = I; a basic result of linear algebra is that AA−1 = I −k −1 k we define negative powers of A via A = A
Matrix Operations
2–12
Examples example 1:
example 2:
1 −1 1 2
−1
1 −1 −2 2
a b c d
=
1 3
2 1 −1 1
(you should check this!)
does not have an inverse; let’s see why:
1 −1 −2 2
=
a − 2b −a + 2b c − 2d −c + 2d
=
1 0 0 1
. . . but you can’t have a − 2b = 1 and −a + 2b = 0
Matrix Operations
2–13
Properties of inverse −1 −1
• A = A, i.e., inverse of inverse is original matrix (assuming A is invertible) • (AB)−1 = B −1A−1 (assuming A, B are invertible) −1 T T −1 (assuming A is invertible) = A • A
• I −1 = I
• (αA)−1 = (1/α)A−1 (assuming A invertible, α 6= 0) • if y = Ax, where x ∈ Rn and A is invertible, then x = A−1y: A−1y = A−1Ax = Ix = x
Matrix Operations
2–14
Inverse of 2 × 2 matrix it’s useful to know the general formula for the inverse of a 2 × 2 matrix:
a b c d
−1
1 = ad − bc
d −b −c a
provided ad − bc 6= 0 (if ad − bc = 0, the matrix is singular) there are similar, but much more complicated, formulas for the inverse of larger square matrices, but the formulas are rarely used
Matrix Operations
2–15
